Robert and Casella (2013) gives the following algorithm,
while Liu, Jun S. (2008) introduces two types of Gibbs sampling strategy.
It is easy to implement this sampler:
## Julia program for Bivariate Gibbs sampler## date: 2018-08-22function bigibbs(T, rho)x = ones(T+1)y = ones(T+1)for t = 1:Tx[t+1] = randn() * sqrt(1-rho^2) + rho*y[t]y[t+1] = randn() * sqrt(1-rho^2) + rho*x[t+1]endreturn x, yend## examplebigibbs(100, 0.5)
We can use the following Julia program to implement this algorithm.
## Julia program for Truncated normal distribution## date: 2018-08-22# Truncated normal distributionfunction rtrunormal(T, mu, sigma, mu_down)x = ones(T)z = ones(T+1)# set initial value of zz = rand()if mu < mu_downz = z * exp(-0.5 * (mu - mu_down)^2 / sigma^2)endfor t = 1:Tx[t] = rand() * (mu - mu_down + sqrt(-2*sigma^2*log(z[t]))) + mu_downz[t+1] = rand() * exp(-(x[t] - mu)^2/(2*sigma^2))endreturn(x)end## examplertrunormal(1000, 1.0, 1.0, 1.2)
Robert and Casella (2013) introduces the following slice sampler algorithm,
and Liu, Jun S. (2008) also presents the slice sampler with slightly different expression:
In my opinion, we can illustrate this algorithm with one dimensioanl case. Suppose we want to sample from normal distribution (or uniform distribution), we can sample uniformly from the region encolsed by the coordinate axis and the density function, that is a bell shape (or a square).
Consider the normal distribution as an instance.
It is also easy to write the following Julia program.
## Julia program for Slice sampler## date: 2018-08-22function rnorm_slice(T)x = ones(T+1)w = ones(T+1)for t = 1:Tw[t+1] = rand() * exp(-1.0 * x[t]^2/2)x[t+1] = rand() * 2 * sqrt(-2*log(w[t+1])) - sqrt(-2*log(w[t+1]))endreturn x[2:end]end## examplernorm_slice(100)
A special case of Completion Gibbs Sampler.
Let's illustrate the scheme with grouped counting data.
And we can obtain the following algorithm,
But it seems to be not obvious to derive the above algorithm, so I wrote some more details
Liu, Jun S. (2008) also presents the DA algorithm which based on Bayesian missing data problem.
Then he argues that copies of in each iteration is not really necessary. And briefly summary the DA algorithm:
It seems to agree with the algorithm presented by Robert and Casella (2013).
It seems that we do not need to derive the explicit form of , if we can directly obtain the conditional distribution. We can use the following Julia program to sample.
## Julia program for Grouped Multinomial Data (Ex. 7.2.3)## date: 2018-08-26# call gamma function#using SpecialFunctions# sample from Dirichlet distributionsusing Distributionsfunction gmulti(T, x, a, b, alpha1 = 0.5, alpha2 = 0.5, alpha3 = 0.5)z = ones(T+1, size(x, 1)-1) # initial z satisfy `z <= x`mu = ones(T+1)eta = ones(T+1)for t = 1:T# sample from g_1(theta | y)dir = Dirichlet([z[t, 1] + z[t, 2] + alpha1, z[t, 3] + z[t, 4] + alpha2, x + alpha3])sample = rand(dir, 1)mu[t+1] = sampleeta[t+1] = sample# sample from g_2(z | x, theta)for i = 1:2bi = Binomial(x[i], a[i]*mu[t+1]/(a[i]*mu[t+1]+b[i]))z[t+1, i] = rand(bi, 1)endfor i = 3:4bi = Binomial(x[i], a[i]*eta[t+1]/(a[i]*eta[t+1]+b[i]))z[t+1, i] = rand(bi, 1)endendreturn mu, etaend# example## dataa = [0.06, 0.14, 0.11, 0.09];b = [0.17, 0.24, 0.19, 0.20];x = [9, 15, 12, 7, 8];gmulti(100, x, a, b)
Let us illuatrate this algorithm with the following example.